265 research outputs found
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
Embedding Stacked Polytopes on a Polynomial-Size Grid
A stacking operation adds a -simplex on top of a facet of a simplicial
-polytope while maintaining the convexity of the polytope. A stacked
-polytope is a polytope that is obtained from a -simplex and a series of
stacking operations. We show that for a fixed every stacked -polytope
with vertices can be realized with nonnegative integer coordinates. The
coordinates are bounded by , except for one axis, where the
coordinates are bounded by . The described realization can be
computed with an easy algorithm.
The realization of the polytopes is obtained with a lifting technique which
produces an embedding on a large grid. We establish a rounding scheme that
places the vertices on a sparser grid, while maintaining the convexity of the
embedding.Comment: 22 pages, 10 Figure
Strictly convex drawings of planar graphs
Every three-connected planar graph with n vertices has a drawing on an O(n^2)
x O(n^2) grid in which all faces are strictly convex polygons. These drawings
are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids.
More generally, a strictly convex drawing exists on a grid of size O(W) x
O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds
are obtained when the faces have fewer sides.
In the proof, we derive an explicit lower bound on the number of primitive
vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The
revision includes numerous small additions, corrections, and improvements, in
particular: - a discussion of the constants in the O-notation, after the
statement of thm.1. - a different set-up and clarification of the case
distinction for Lemma
Embedding Stacked Polytopes on a Polynomial-Size Grid
We show how to realize a stacked 3D polytope (formed
by repeatedly stacking a tetrahedron onto a triangular
face) by a strictly convex embedding with its n vertices
on an integer grid of size O(n4) x O(n4) x O(n18). We
use a perturbation technique to construct an integral 2D
embedding that lifts to a small 3D polytope, all in linear
time. This result solves a question posed by G unter M.
Ziegler, and is the rst nontrivial subexponential upper
bound on the long-standing open question of the
grid size necessary to embed arbitrary convex polyhedra,
that is, about effcient versions of Steinitz's 1916
theorem. An immediate consequence of our result is
that O(log n)-bit coordinates suffice for a greedy routing
strategy in planar 3-trees.Deutsche Forschungsgemeinschaft (DFG) (Grant No. SCHU 2458/1-1
Convexity-Increasing Morphs of Planar Graphs
We study the problem of convexifying drawings of planar graphs. Given any
planar straight-line drawing of an internally 3-connected graph, we show how to
morph the drawing to one with strictly convex faces while maintaining planarity
at all times. Our morph is convexity-increasing, meaning that once an angle is
convex, it remains convex. We give an efficient algorithm that constructs such
a morph as a composition of a linear number of steps where each step either
moves vertices along horizontal lines or moves vertices along vertical lines.
Moreover, we show that a linear number of steps is worst-case optimal.
To obtain our result, we use a well-known technique by Hong and Nagamochi for
finding redrawings with convex faces while preserving y-coordinates. Using a
variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and
Nagamochi's result which comes with a better running time. This is of
independent interest, as Hong and Nagamochi's technique serves as a building
block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201
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Discrete Geometry
The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants
A Quantitative Steinitz Theorem for Plane Triangulations
We give a new proof of Steinitz's classical theorem in the case of plane
triangulations, which allows us to obtain a new general bound on the grid size
of the simplicial polytope realizing a given triangulation, subexponential in a
number of special cases.
Formally, we prove that every plane triangulation with vertices can
be embedded in in such a way that it is the vertical projection
of a convex polyhedral surface. We show that the vertices of this surface may
be placed in a integer grid, where and denotes the shedding diameter of , a
quantity defined in the paper.Comment: 25 pages, 6 postscript figure
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